How to batch export shapekeys as OBJ from the active object?
This script will iterate over the active object's shape keys, set each in turn to 1.0 (while the others are set to 0.0) and save this as an OBJ file.

import bpy
from os.path import join

o = bpy.context.object # Reference the active object

# Change this to the actual folder you want to save your OBJ files in
exportPath = "C:/tmp/obj" 

# Reset all shape keys to 0 (skipping the Basis shape on index 0
for skblock in o.data.shape_keys.key_blocks[1:]:
    skblock.value = 0

# Iterate over shape key blocks and save each as an OBJ file
for skblock in o.data.shape_keys.key_blocks[1:]:
    skblock.value = 1.0  # Set shape key value to max

    # Set OBJ file path and Export OBJ
    objFileName = skblock.name + ".obj" # File name = shapekey name
    objPath = join( exportPath, objFileName )
    bpy.ops.export_scene.obj( filepath = objPath, use_selection = True )

    skblock.value = 0 # Reset shape key value to 0